Why isn’t there a gravitational pull on fire?

Carry a helium or hydrogen filled balloon in a car, and accelerate the car. (Be careful; this can be dangerous.) The balloon will not be pulled to the rear of the car; it will move to the front! (That's why I said you have to be careful.)

The balloon is, of course, pushed backwards by the acceleration (according to the equivalence principle, this is equivalent to rearward gravity), but the air, since it weighs more per volume, is pushed more. So it is what goes backwards, while the balloon is forced forward by the heavier flowing air.

Likewise from gravity. A balloon goes up because the heavier air falls down around it, displacing the balloon. And similarly for the hot plasma known as fire. It weighs less per cubic centimeter than does the air around it, so the air drops and pushes the plasma up.